Ivanovic's battles with her emotions were there for all to see and Peer was able to seize the match with a consistent second set.
Peer broke again at 4-4 and then closed out the match with Ivanovic's long shot on match point summing up her recent misery. Peer will now play German Angelique Kerber who beat India's Sania Mirza comfortably 6-4, 6-1.
Lucky loser Jesse Levine and Chile's Nicolas Massu lost four-set thrillers in the opening round of the men's singles.
Israel No. 1 Dudi Sela is in action on Tuesday.
Full details in Friday's JC.